Question

In the relation $y=a\,\,\cos \left( \omega t-kx \right)$ ,the dimensional formula for $k$ is :

• A. $\left[ {{M}^{0}}{{L}^{-1}}{{T}^{-1}} \right]$
• B. $\left[ {{M}^{0}}L{{T}^{-1}} \right]$
• C. $\left[ {{M}^{0}}{{L}^{-1}}{{T}^{0}} \right]$
• D. $\left[ {{M}^{0}}LT \right]$

Answer 1

Answer: (C)

$\left[ {{M}^{0}}{{L}^{-1}}{{T}^{0}} \right]$

Answer 2

Every equation relating physical quantities should be in dimensional balance.
The given equation is in dimensional balance, hence the dimensions of the terms on both sides of the equation must be the same.
$\therefore y=a \cos (\omega t-k x)$
$y$ has dimensions of length and a that is amplitude also has dimensions of length, hence $(\omega t-k x)$ should be dimensions, that is
$[k]=\frac{1}{[x]}=\frac{1}{[L]}$
Dimensions of $k=\left[M^{0} L^{-1} T^{0}\right]$

Therefore, the correct option is (C): $\left[ {{M}^{0}}{{L}^{-1}}{{T}^{0}} \right]$